Math 3760: Topics in Topology

General References

• Differential Topology, V. Gullemin and A. Pollack. "Friendly" and a nice read.
• Foundations of Differentiable Manifolds and Lie Groups, F. Warner. Less friendly. Terse to a fault and lacking
  examples in places, it also does everything "right" (a double-edged sword, but very useful for referencing).
• Algebraic Topology, Allen Hatcher. The go-to algebraic topology reference for the non-algebraic topologist.
  It is freely available on his website (and linked above).
• The standard reference for virtually anything in PL topology is Rourke & Sanderson's Introduction to
  Piecewise-Linear Topology. More accessible (also less general and less formal) is M.A. Armstrong's Basic
  Topology, which also has lots of other good stuff.

Project Possibilities

• For the classification of topological 1-manifolds:
  The Classification of 1-Manifolds: a Take-Home Exam. David Gale, The American Mathematical Monthly.
• For the classification of smooth (compact) 1-manifolds: Appendix 2 of Guillemin & Pollack.
• For existence and uniqueness of smooth structures on surfaces:
  The Kirby torus trick for surfaces. Allen Hatcher, arXiv:1312.3518.
• The first "exotic'' smooth structures, on the 7-sphere:
  On Manifolds Homeomorphic to the 7-sphere. J. Milnor, Annals of Mathematics, 1956.
• There is a nice explanation of the Jordan separation theorem, the Jordan curve theorem, and the Schoenflies
  theorem in Munkres' Topology. For an algebraic topology proof of the higher-dimensional Jordan separation
  theorem, and an elementary construction of the Alexander horned sphere, see Appendix 2.B of Hatcher's
  Algebraic Topology.
• A proof of the generalized Schoenflies theorem, M. Brown. Bulletin of the AMS, 1960.
  The title says it all. I have not read this proof but it is amazingly short (3 pp) and self-contained (1 reference).

Further Reading

• Boy's surface (at Wikipedia): an immersion of RP^2, the real projective plane, into R^3.
• Lectures on Diffeomorphism Groups of Manifolds. Alexander Kupers.
  There's a whole lot here, with heavy machinery, but the beginning is nicely written and approachable.
• Groups of Homotopy Spheres: I. M. Kervaire and J. Milnor, Annals of Mathematics, 1962.
  Describes a group structure (with operation connected sum) on the set of exotic spheres up to
  diffeomorphism in dimensions at least 5 (after the h-cobordism theorem).
• The math review (subscription needed) of J. Cerf's proof (in a series of four substantial papers written
  in French) that "\Gamma_4=0", ie. that any two OP diffeomorphisms of S^3 are smoothly isotopic.
• Stable homeomorphisms and the annulus conjecture. R. Kirby, Annals of Mathematics, 1969.
  A proof of the Stable Homeomorphism Conjecture, hence also the Annulus Conjecture, in dimensions
  at least 5. This implies by standard arguments that any two OP homeomorphisms of S^n are isotopic.
• Extending Diffeomorphisms, R. Palais. Proceedings of the AMS, 1960.
  Theorem B here implies that the connected sum of smooth manifolds is well defined. The corresponding
  result for topological manifolds uses the Annulus Conjecture.
• The Hauptvermutung Book, ed. A.A.Ranicki.
  Actually a collection of papers disproving the "hauptvermutung der kombinatorischen topologie", on
  homeomorphic simplicial complexes being isomorphic after subdivision. Ranicki collected them and
  wrote a nice introduction stating the problem and giving context.